Ulam’s spirals reveal the regularity of prime numbers

In general, the diagonals in an Ulam spiral can be described by the first formula. Obviously, the function does not work with any arbitrary set of a B C And n (which all contain integer values) Returns a prime number. If that were the case, one would have solved one of the biggest unanswered questions in mathematics: that of the distribution of prime numbers. However, the Spiral of Ulam shows that there are certain groups of far And c There, for the one with the job and (n) It can calculate many more prime numbers than other combinations.

Ulam found a few of these combinations himself; Other mathematicians also frequently dealt with Ulam spirals (as they are now called). However, there is still no definitive explanation for the pattern in the spirals.

Stanisław Ulam was not the first to come across a phenomenon of this kind. As early as 1932, American herpetologist Lawrence Klauber arranged the natural numbers as a triangle, with 1 at the top. If you mark prime numbers there, they are most often on diagonal and vertical lines. Herpetology, by the way, is not a mathematical subject, but the science of amphibians and reptiles – Klopper was one of the leading experts on rattlesnakes. Not necessarily the kind of research most likely to yield discoveries about prime numbers, but creativity is a bit like love in this respect: it sees, to paraphrase Shakespeare, with the mind rather than the eyes—and meets mathematicians and rattlesnake researchers alike.